scholarly journals Unifying the Dynkin and Lebesgue–Stieltjes formulae

2017 ◽  
Vol 54 (1) ◽  
pp. 252-266 ◽  
Author(s):  
Offer Kella ◽  
Marc Yor
Keyword(s):  
Bounded Variation ◽  
Sufficient Conditions ◽  
Jump Diffusion ◽  
Continuous Functions ◽  
Local Martingale ◽  
Functions Of Bounded Variation ◽  
Deterministic Function ◽  
Stieltjes Integration ◽  
Change Of Variable ◽  
Martingale Convergence

AbstractWe establish a local martingaleMassociate withf(X,Y) under some restrictions onf, whereYis a process of bounded variation (on compact intervals) and eitherXis a jump diffusion (a special case being a Lévy process) orXis some general (càdlàg metric-space valued) Markov process. In the latter case,fis restricted to the formf(x,y)=∑k=1Kξk(x)ηk(y). This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue–Stieltjes integration (change of variable) formula for (right-continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed, then this local martingale becomes anL2-martingale. Convergence of the product of this Martingale with some deterministic function ( of time ) to 0 both inL2and almost sure is also considered and sufficient conditions for functions for which this happens are identified.

Curvature-dependent energies: a geometric and analytical approach

2017 ◽  
Vol 147 (3) ◽  
pp. 449-503 ◽  
Author(s):  
Emilio Acerbi ◽  
Domenico Mucci
Keyword(s):  
Bounded Variation ◽  
Analytical Approach ◽  
Total Curvature ◽  
Representation Formula ◽  
Energy Functional ◽  
Continuous Functions ◽  
Explicit Representation ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Relaxed Energy

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

On the Correctness of Nonlinear Boundary Value Problems for Systems of Generalized Ordinary Differential Equations

1996 ◽  
Vol 3 (6) ◽  
pp. 501-524
Author(s):  
M. Ashordia
Keyword(s):  
Bounded Variation ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Continuous Operator ◽  
Functions Of Bounded Variation ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
Carathéodory Class ◽  
Generalized Ordinary Differential Equations

Abstract The concept of a strongly isolated solution of the nonlinear boundary value problem dx(t) = dA(t) · f (t, x(t)), h(x) = 0, is introduced, where A : [a, b] → Rn×n is a matrix-function of bounded variation, f : [a, b] × Rn → Rn is a vector-function belonging to a Carathéodory class, and h is a continuous operator from the space of n-dimensional vector-functions of bounded variation into Rn . It is stated that the problems with strongly isolated solutions are correct. Sufficient conditions for the correctness of these problems are given.

DEFINITION AND CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750048 ◽  
Author(s):  
Y. S. LIANG
Keyword(s):  
Fractal Dimension ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Nondifferentiable Functions ◽  
Differentiable Functions ◽  
Closed Intervals ◽  
Bounded Variation Functions

The present paper mainly investigates the definition and classification of one-dimensional continuous functions on closed intervals. Continuous functions can be classified as differentiable functions and nondifferentiable functions. All differentiable functions are of bounded variation. Nondifferentiable functions are composed of bounded variation functions and unbounded variation functions. Fractal dimension of all bounded variation continuous functions is 1. One-dimensional unbounded variation continuous functions may have finite unbounded variation points or infinite unbounded variation points. Number of unbounded variation points of one-dimensional unbounded variation continuous functions maybe infinite and countable or uncountable. Certain examples of different one-dimensional continuous functions have been given in this paper. Thus, one-dimensional continuous functions are composed of differentiable functions, nondifferentiable continuous functions of bounded variation, continuous functions with finite unbounded variation points, continuous functions with infinite but countable unbounded variation points and continuous functions with uncountable unbounded variation points. In the end of the paper, we give an example of one-dimensional continuous function which is of unbounded variation everywhere.

Approximation of functions of bounded variation by integrated Meyer-König and Zeller operators of finite type

2012 ◽  
Vol 49 (2) ◽  
pp. 254-268
Author(s):  
Tiberiu Trif
Keyword(s):  
Rate Of Convergence ◽  
Finite Type ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
Approximation Of Functions ◽  
Durrmeyer Operators

I. Gavrea and T. Trif [Rend. Circ. Mat. Palermo (2) Suppl. 76 (2005), 375–394] introduced a class of Meyer-König-Zeller-Durrmeyer operators “of finite type” and investigated the rate of convergence of these operators for continuous functions. In the present paper we study the approximation of functions of bounded variation by means of these operators.

scholarly journals Trapezoid type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation

2020 ◽  
Vol 12 (1) ◽  
pp. 30-53
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Bounded Variation ◽  
Continuous Functions ◽  
Fractional Integrals ◽  
Functions Of Bounded Variation ◽  
Hölder Continuous ◽  
Hölder Continuous Functions ◽  
Hadamard Fractional Integrals

AbstractIn this paper we establish some trapezoid type inequalities for the Riemann-Liouville fractional integrals of functions of bounded variation and of Hölder continuous functions. Applications for the g-mean of two numbers are provided as well. Some particular cases for Hadamard fractional integrals are also provided.

The form of locally defined operators in waterman spaces

2021 ◽  
Vol 71 (6) ◽  
pp. 1529-1544
Author(s):  
Małgorzata Wróbel
Keyword(s):  
Banach Spaces ◽  
Bounded Variation ◽  
Composition Operators ◽  
Representation Formula ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
Spaces Of Continuous Functions ◽  
Local Operators ◽  
Uniformly Bounded

Abstract A representation formula for locally defined operators acting between Banach spaces of continuous functions of bounded variation in the Waterman sense is presented. Moreover, the Nemytskij composition operators will be investigated and some consequences for locally bounded as well as uniformly bounded local operators will be given.

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